What is Inverse Kinematics?
What is Inverse Kinematics?

There are two main ways to pose a chain of bones:

"Forward Kinematics" (FK) is the normal way of manipulating a bone chain, based on parent-child relationships. You need to rotate each bone individually. 

Inverse Kinematics (IK) is another way, where the movement of the chain is determined by a "target" bone (and optionally a "pole target" bone) instead of the normal parent-child relationships. Basically you position the end of the chain, and the IK solver calculates what the rest of the bones need to do in order to make that happen.

Why do we need the Inverse Kinematic constraint in rigging? 

We don't need it, it just makes life easier. Often much easier.

When should I use it?

In every situation where you need the end of the bone chain to be in a specific place - like characters grabbing something, or the foot on the floor while walking. 

What does it do to the armature?

Rotates the bones of the chain into position according to the two control bones (target, pole).